In-Quest-of-the-Philosophers--Stone-Nonlinearities

InQuestofthePhilosophers’Stone:NonlinearitiesandVolatilityinFinancialSeriesRenzoG.AvesaniDip.diSc.EconomicheUniv.diBresciaViaPorcellaga2125121Brescia,ItalyLuciaBuzzigoliDip.diStatistica\G.ParentiUniv.diFirenzeVialeMorgagni5950134Firenze,[e-mail]\G.ParentiUniv.diFirenzeVialeMorgagni5950134Firenze,[e-mail]{March1996ThanksareduetoB.Chiandotto,C.Giannini,R.Orsi,M.Salmon,andI.Viscofortheircom-ments,andtoBarbaraPaciniforherhelpinintroducingustothenon{andsemi{parametricworld.FinancialsupportfromtheMURSTandCNRisgratefullyacknowledged.1Contents1Introduction32ModelsfortheConditionalVariance92.1TheOriginalModel::::::::::::::::::::::::::112.2GARCH,IGARCHandGARCH-MModels:::::::::::142.3AFirstClassofExtensions:::::::::::::::::::::152.4ThePathtoMoreGeneralFormulations:::::::::::::182.5TheSecondGenerationModels:::::::::::::::::::192.6ModellingAsymmetricPatterns::::::::::::::::::202.6.1EGARCHModels:::::::::::::::::::::::202.6.2TGARCHModels:::::::::::::::::::::::222.6.3Alternativeproposals:::::::::::::::::::::232.7OtherExtensions::::::::::::::::::::::::::::262.8SomeConnections:::::::::::::::::::::::::::292.8.1BilinearProcesses:::::::::::::::::::::::312.8.2ARCHandBilinearProcesses:::::::::::::::333DerivativeApplications353.1OptionPricing:::::::::::::::::::::::::::::353.1.1TheOriginalBlackandScholesModel::::::::::353.1.2JumpRisk::::::::::::::::::::::::::::383.1.3StochasticVolatility::::::::::::::::::::::413.2InterestRatesTermStructure:::::::::::::::::::443.3ArbitrageBasedSolutionStrategies::::::::::::::::464Risk{RelatedMeasures485BlackBoxesandMoneyMachines515.1NeuralNetworks::::::::::::::::::::::::::::515.2TradingStrategies:::::::::::::::::::::::::::566SomeNon-concludingRemarks597TechnicalAppendix6221IntroductionAsurveyonTheEconomist(Oct.9th,1993)wasrecentlydedicatedtothefrontiersofnance,avirtualplacewherenancialanalysts,statisticians,mathematicians,economists,computerscientistsandengineersmeettodevisenewtoolstostudythebehaviorofnancialmarkets.Therelationshipbetweenacademicresearchandoperationalactivityisadicultoneinthiseld.Aturningpointinthisrelationship,andinthestyleofnancialmodellingtookplaceasaconsequenceofthecrashofinOctober1987,alsoknownastheendofnancialinnocence,aneventwhichmorethananyotherinrecenttimesalteredtheprevailingcreedofthenancialcommunity:linearityandrationality.Thediusion(andthesuccess)ofnonlinearmethodsofanalysisinthelastfewyearshasbeensowidespreadastowarrantalargeinvestmentintheareabynancial3rms.Computer{basedtradingandfundmanagementseemtobefast{growingareasofactivity:theobviousremarkisthatifitkeepsongoingitmustmeanthatthereisprotbehindit,thebestperformancediagnostics.Infact,secrecyaboutmethodolo-giesandresultsaccompaniessuccess;atmost,onehearsrumorsaboutastronomicalguresgivenasbonusestothePh.D.inmathematicsworkinginthebasementoffamoustradingcompanies.Ontheotherhand,aprofusionofpapersaccompaniesagoodexplanatoryperformancein{samplebutadisappointingoneforforecasting.Itsucestoquoteacommentoftenheardatacademicseminars,\Ifitworkedinforecasting,Iwouldnotbehere.Moreover,onthepartoftheoperatingunitsofnancialcompanies,thereisagenerallackofinterest,ifnotamistrust,forstatisticalproperties,asymptoticresultsandsoon:theinterestisfocusedonthenumbersthatatoolcanproduce.Formanyatraderablack{boxinthebackoceisallthatsuf-ces:rather,whatsucesisthatpushingabutton,itprovidesthemwiththerightsuggestions.Inthissurvey,wedonotintendtoandcannotproposeasurveyonallthetheor-iesandasummaryoftheempiricalresultsobtainedintherecentyears.Thepapergatherssomereectionsandrecallssomeresultstoprovidethestageforfurtherre-ectionsanddiscussions.Weleftoutanumberofimportanttopics,mainlybecausethepaperwouldhaveleavenedbeyondreadability.Thus,nonlinearmodelsfortheexpectedvalueofaprocessconditionalonaninformationset,thetestsfornonlinear-ity,themodelsof(low{dimensional)deterministicchaoswhichinrecenttimeshavebeenstudiedtoproposeapredictablealternativetononlinearstochasticmodels(cf.Brock,HsiehandLeBaron,1991;DeGrauwe,DewachterandEmbrechts,1993),orthestudiesexploitingtheavailabilityofdataatveryhighfrequencies(GoodhartandFigliuoli,1991;Goodhartetal.,1993)willnotbedealtwithhereindetail,inspiteoftheirrelevancefortheissuesathand.Thedebateabouttheanalysisofnancialtime{seriesrevolvesaroundthecausesofthewidespreadbeliefthatthemarketeciencyhypothesisshouldbeburied:itisrecognizedthatlinearmodelsareinadequatetounravelagents’decisionsastheyarereectedinanassetprice.Theowofinformation(virtuallycontinuous),thedierent(andpossiblyasymmetric)reactionsbyagentstothearrivalofnews,theexistenceofatimescaleforagentsdierentfromcalendartime(theso-calledtimedeformationeect),theinteractionofpolicyactionsandtheformationofexpectationsinthemarketsareallelementsthatmayaccountforthefailureoflinearmodelstoexplainassetreturns.AcaseinpointisthementionedsurprisecausedbytheOctober1987crashwhichupsetinmanywaysthenancialanalysts,notonlyfor